To rule out the ice maker:
Flip the bar to the up position as if the ice tray is full. Leave it that way for 24 hours and monitor. Next empty the ice tray and watch it go. If your ice maker is a component of Sense, this should help you to confirm it.
If you are pretty sure the 11 seconds are coming from your HVAC gear, use the timing pattern and your ears. If there is an ignition cycle occurring to fire up the boiler, or air quality testing, you can probably hear it click. Try to predict when it will happen and go down there. Listen for it.
If someone with a bigger brain than me could review the power schematics here:fieldcontrols.com

We might be able to zero in on the Gas Pressure switch , or the direct Spark ignition module.
They might think your nuts, but you could call them, Field Controls Technical Support at 1-800-742-8368
I bet the Can in a fan stuff is always running or trying to determine if it needs to be running

Looking at this, could it be one or more of your Nests? Wouldn’t they draw power from the boiler? (Assuming the controller is in the Boiler/ Boiler room) Here’s my logic:
After leaving the circuit off for a while, you saw a change in behavior, a longer power draw than seen before. This mirrors something that happened in my house as I was trying to locate phantom draw. After turning off a circuit I had a much larger and longer power draw when I energized it again. Traced it to the security alarm recharging the backup battery. This mirrors what you saw while trying to track this item down. Nests have a built in battery that should regularly stay charged, but I can imagine instances where a failed or failing battery may periodically pull a draw to maintain a particular float voltage. That, or perhaps there is a regular task that runs every so often that requires a stronger current draw (reporting to the cloud or such).

If this is still going on, I’d consider unplugging one or all of the nests and see how the behavior changes.

This is a really good theory. I am going to check that out. It’s easy to predict as it happens the same time on the hour. I’ve tried listening for something in the boiler room and haven’t heard a pin drop…

Have you had any luck? I have sort of the same thing… it cycles pretty consistently… I’ve tried unplugging the fridge, which doesn’t affect the pattern as well as turning the boiler breaker off… Sense said it was a Fridge… but the program still saw the current draw when I unplugged my real fridge…,

It is definitely something with a motor (pump, compressor). This pattern is classically seen with motors. Overcoming inertia (locked rotor amps) takes a strong current inrush, but once that’s done, current consumption drops.